Pertubation and 3rd degree Mandelbrot

[knighty]

So you did 7th degree by hand!  Impressive! and looks good. 

[cKleinhuis]

lol, dudes, very amazing work, any chance for creatung pertubation optimization in an "automated" way !??!
i think i comes down to symbolic problem sovling .... we all know computers can do such, although it might be a little frightening, but isnt that we made computers for !?
so, perhaps ... formulate it in a mathematica kind of way, in a way that it can be solved automatically for any (lol, non integer, non real  ) exponent

and, dudes, you know why i am askin ? it is just because hybridisation techniques like alternations, could really really really be enlighted by that, because we have not yet deeply zoomed the easiest of the pertubations, like alternating a 2 and a 3 exponent mandelbrot together ...

just dreaming ahead .... those damn computer slaves are made for such things, arent they !?

[knighty]

That's what I'm still dreaming of too. Not only for 2D escape time fractals but also for 3D ones. Yes, it possible... but difficult to code for one person alone . That would be a good collaborative work.
IMHO, the main issue is mantaining good performance while having the ability to set custom formulas. AFAIK, LLVM is the only tool that is powerful enought for the job but it's heavy, very heavy. 
Another solution would be to use plugins just like mandelbulb3D or apophysis.

@Kalles Fraktaler:
In case it will be useful for series approximation for higher power MB sets, here is the "general" formula of the derivatives (p>=3):





It becomes more and more cumbersome for higher derivatives (there is a structure though).

[Kalles Fraktaler]

Thanks knighty, I will examine these to see if they can yield more iterations.

Formula interpretation and derivation... yeah, that would be something 

I received an interesting suggestion to implement PSA on the formula X_n+1=X_n^(1/X_n)+X₀
But I currently don't know how to make fractional power of complex number or derivative them...

[Kalles Fraktaler]

Unfortunately no more iterations could be skipped using 3 terms on 4th power Mandelbrot, compared to using 1 term.
Thanks anyway!

[therror]

Has anyone used perturbation theory for arbitrary (non-integer) power?

[Kalles Fraktaler]

Has anyone used perturbation theory for arbitrary (non-integer) power?

Has anyone implemented arbitrary precision at all for arbitrary non-integer power?
To calculate power on complex numbers since and cosine is involved, and for arbitrary power, Taylor series would be needed, which involves iterations. Iterations on iterations. I imagine it would be incredible slow?

That's what I'm still dreaming of too. Not only for 2D escape time fractals but also for 3D ones. Yes, it possible... but difficult to code for one person alone . That would be a good collaborative work.
IMHO, the main issue is mantaining good performance while having the ability to set custom formulas. AFAIK, LLVM is the only tool that is powerful enought for the job but it's heavy, very heavy. 
Another solution would be to use plugins just like mandelbulb3D or apophysis.

@Kalles Fraktaler:
In case it will be useful for series approximation for higher power MB sets, here is the "general" formula of the derivatives (p>=3):
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It becomes more and more cumbersome for higher derivatives (there is a structure though).

Knighty, your general formula works like a charm.
Would it be possible to make arbitrary number of terms as well?